First, your assertion that Warlord healing equating to shouting wounds closed being edition war rhetoric is true if and only if you play D&D the way Tony plays.
'Shouting wounds closed is /only/ edition war rhetoric' is absolutely true on two counts: 1) it was quite literally used as rhetoric in the edition war and never used before that, and 2) it was factually false under the highly abstract way hit points were handled in 4e (and all the way back to 1e AD&D, and still in 5e, for that matter).
Now, if you change the way hps work in D&D, you can /make/ the Warlord's form of hp-restoration 'incoherent,' as you clearly demonstrate. But, first you have to change a definition of hps that's been with the game since 1979.
It's one of the many things about the rhetoric of the edition war that was, well, incoherent.
The delineation between what we are and aren't okay with when it comes to healing confuses the bejezus out of me...
If HP are all 'meat' and only magic can instantaneously heal, then are you allowing short rest HD healing?
Reasonably, yes. Under a variant in which hps were re-defined that way, eliminating a number of things would make sense. HD would be among them. So would second wind, Vicious Mockery (and the 'psychic' damage type, in general), overnight healing, and things like Healer's Kits and the Healer feat would at least need some re-writing.
5e empowers DMs to do just those sorts of things if they want to. That's why it can afford to include things like HD, even in the Standard Game.
This opens up yet another branch of discussion/argument.
Forcing (or at least trying to force) someone you're in melee with to move is just fine - push the orc backwards against the wall, or throw it sideways into the mud, or drag it towards you and away from its allies, etc.
Don't forget forcing an enemy back or giving ground as a way of moving them around. It's not physically pushing or grabbing them, but can still get them where you want them...
Where problems arise is when you're non-magically forcing someone else, not within your physical reach, to move. Sure you can *suggest* your ally move to a better position or yell at the left-flank fighter to shove her foe into your reach, but it should be up to them-as-characters whether or not they follow those suggestions (or orders)
In 4e, that was modeled by making the granted action a free action for the targeted ally - they could always choose not to take the free action. 5e's Reaction is much more 'expensive' in the Action Economy, so any ability that required an ally to use it would have to be that much more potent to be balanced, but it's still a workable idea.
And non-magically forcing enemies to move at a distance makes no sense at all.
It's not called 'forced' or 'involuntary' movement in 5e anymore, so hopefully that misunderstanding will be less of an issue.
The business equation boils down to:
who will quit over it being added versus who will finally buy in if it is...
Anyone that appalled by things like non-magical healing saw Second Wind and HD and never gave 5e a chance. Fans who want the Warlord added to 5e /are/ fans of 5e, and there's nothing wrong with 5e fans wanting something cool added to the game. There's a long list of things that haven't made it into 5e yet. Psionics and the Warlord were two of the most noticeable in their absence from the PH, and we've already got Psionics on track.
No current fan of 5e will go ballistic and rage-quit over a new class, module, or other opt-in addition to the Advanced Game. Anyone that uptight about the game not being played exactly the same way at every table has already been driven away screaming by the DMG, not to mention the level of DM empowerment inherent in the core mechanics.
Fans who feel very proprietary about the Standard Game are probably more concerned with the re-designing of the Ranger.